It's very odd to me because many people have said that switching between coins in Ethash should be the same.
Maybe ETC takes up more power?
It shouldn't. Maybe dual mining with claymore is working somehow different for ETC+DCR than for ETH+DCR. Though very unlikely.
One crucial thing missing from your description — does ETH+DCR work now? You said you were mining ETH+DCR for 2 weeks without any problems. Then you switched to ETC+DCR recently, and you're having problems with the rig restarting. Did you try going back to ETH+DCR? Cause if you didn't, then there's a chance that ETH+DCR isn't stable anymore as well and it just could be something not related to ETC in any way.